Exercise 2084Combinatorics › Binomial coefficientsM1Added 3 Nov 2026

Chu–Vandermonde for negative upper index

#Chu-Vandermonde#generalized binomial#identity

Statement

State the Chu–Vandermonde identity and compute $\sum_{k=0}^n\binom{-1}{k}\binom{2}{n-k}$ .

Hint

Chu–Vandermonde: $\sum_k\binom{r}{k}\binom{s}{n-k}=\binom{r+s}{n}$ for arbitrary complex $r,s$ .

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Identity. $\sum_{k=0}^n\binom{r}{k}\binom{s}{n-k}=\binom{r+s}{n}$ .

Compute. $r=-1$ , $s=2$ . Sum $=\binom{1}{n}$ . For $n=0$ : $1$ . For $n=1$ : $1$ . For $n\ge2$ : $0$ .

Verify. $\binom{-1}{k}=(-1)^k$ . For $n=1$ : $\binom{-1}{0}\binom{2}{1}+\binom{-1}{1}\binom{2}{0}=2-1=1$ . ✓

Conclusion. $\sum_k\binom{-1}{k}\binom{2}{n-k}=\binom{1}{n}$ , i.e. $1$ for $n\in\{0,1\}$ , $0$ otherwise.